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Federal Democratic Republic of Ethiopia

Ministry of Education

Minimum Learning Competencies

Mathematics, Grades 9 to 12

2009

Minimum Learning Competencies Grades 9–12

1

Statement of Minimum Learning Competencies (MLCs) in Mathematics for Grade 9 & 10

Area of Competency Grade 9 Grade 10

I. NUMBER SYSTEM

The real number system

• identify natural numbers and integers

• define prime numbers and composite numbers

• determine common factors and common multiples of pairs of

numbers

• show that repeating decimals are also rational numbers

• identify irrational numbers

• locate some irrational numbers on a number line.

• define real numbers.

• describe the correspondence between real numbers and points

on a numbers line.

• Realize the relationship between a power with fractional

exponent and a radical form.

• Convert powers with fractional exponent to radical form and

vice-versa

• perform any one of the four operation on the set of real

numbers

• use the laws of exponents to simplify expression.

• give appropriate upper and lower bounds for a given data to a

specified accuracy (e.g. rounding off)

• express any positive rational number in its standard form.

• explain the notion of rationalization.

• identify a rationalizing factor for a given expression.

• use the Euclid's division algorithm to express given quotients

of the form

q

p where, p > q.

II. ALGEBRA

Solving Equations

and Inequalities

• Solve equations involving exponents and radicals

• Solve simultaneous equation

• identify the three cases of solutions of simultaneous

equations (a unique solution, no solution, infinitely many

solutions)

• Solve equations involving absolute value

• Solve quadratic equations by using any one of the three

• describe sets using internal notation.

• solve inequalities involving absolute value of

linear expression

• solve system of linear inequalities in two variables

by using graphical method

• solve quadratic inequalities by using product

properties


2


methods

• Apply Viete's theorem to solve related problems

• solve quadratic inequalities using the sign chart

method.

• solve quadratic inequalities using graphs

III. SETS

• describe sets in different ways

• identify the elements of a given set

• explain the notion "empty set" and "universal set"

• determine the numbers of subsets of a given finite set and list

them.

• give the power set of a given set

• determine the number of proper subsets of a given finite set

and list them.

• distinguishes between equal sets and equivalent sets

• find equal sets and equivalent sets to a given set

• determine number of elements in the union of two finite set.

• describe the properties of "union" and "intersection" of sets.

• determine the absolute complement of a given set.

• determine the relative complement of two sets

• determine the symmetric difference of two sets.

• determine the Cartesian product of two sets.

IV. RELATION AND

FUNCTION

• define the notions "relation", "domain" and "range"

• draw graphs of relations

• use graphs of relation to determine domain and range

• define function

• determine the domain and range of a given function.

• determine the sum difference, produced and quotient of

functions.

• Evaluate combination of functions for a given values from

their respective domain.

• sketch graphs of linear functions

• describe the properties of the graphs of linear functions.

• sketch the graphs of a given quadratic function.

• define the polynomial function of one variable

• identify the degree, leading coefficient and

constant terms of a given polynomial functions.

• give different forms of polynomial functions

• perform the four fundamental operation on

polynomials

• state and apply the polynomial division theorem

• state and apply the Factor Theorem

• determine the zero(s) of a given polynomial

function

• state and apply the Location theorem to

approximate the zero(s) of a given polynomial

function

IV. RELATION AND • describe the properties of the graphs of given quadratic • apply the rational root test to determine the zero(s)


3


FUNCTION (cont.)

functions

• determine the maximum and minimum values of a given

quadratic function

of a given polynomial function.

• sketch the graph of a given polynomial function.

• describe the properties of the graphs of a given

polynomial function

• explain what is meant by exponential expression

• state and apply the properties of exponents (where

the exponents are real numbers)

• express what is meant by logarithmic expression

by using the concept of exponential expression

• solve simple logarithmic equation by using the

properties of logarithm

• recognize the advantage of using logarithm to the

base 10 in calculation

• identify the "characteristics" and "mantissa" of a

given common logarithm

• use the table for finding logarithm of a given

positive number and antilogarithm of a number.

• compute using logarithm

• define an exponential function.

• draw the graph of a given exponential function

• describe the graphical relationship of exponential

functions having bases reciprocal to each other

• describe the properties of an exponential function

by using its graph.

• define a logarithmic function

• draw the graph of a given logarithmic function

• describe the properties of a logarithmic function

by using its graph

• describe the graphical relationship of logarithmic

function having bases reciprocal to each other.

• describe how the domains and ranges of y = ax and

y = logax are related

• explain the relationship of the graphs of y = ax

and y = logax


4


IV. RELATION AND

FUNCTION (cont.)

• solve equations involving exponents and

logarithms as well

• solve problems, involving exponential and

logarithmic functions, from real life.

• define the sine, cosine and tangent functions of an

angle in the standard position.

• determine the values of the functions for an angle

in the standard position, given the terminal side of

that angle.

• determine the values of the sine, cosine and

tangent functions for quadrantal angles

• locate negative and positive angles

• determine the values of trigonometric functions

for some negative angles.

• determine the algebraic signs of the sine, cosine

and tangent functions of angles in different

quadrants.

• describe the relationship between trigonometrical

values of complementary angles.

• describe the relationship between trignonometrical

values of supplementary angles.

• determine the relationship between

trigonometrical values of coterminal angles.

• determine the trigonometrical values of large

angles

• construct a table of values for y = sin θ where -2π

≤ θ ≤ 2π.

• draw the graph of y = sin θ

• determine the domain range and period of the sine

function.

• construct a table of values for y = cos θ , where -

2π ≤ θ ≤ 2π.

• draw the graph of y = cos θ

• determine the domain, range and period of the

cosine function.


5


IV. RELATION AND

FUNCTION (cont.)

• construct a table of values for y = tan θ where -2 π

≤ θ ≤ 2π.

• draw the graph the tangent function y = tanθ.

• determine the domain, range and period of the

tangent function.

• discuss the behavior of the graph of tangent

function

• define the cosecant function

• determine the values of cosecant function for some

angles.

• define the secant function.

• determine the values of secant function for some

angles.

• define the cotangent function

• determine the values of cotangent function for

some angles.

• explain the concept of co-functions.

• derive some of the trignometric identities.

• identity the quotient identities.

• solve problems related to trigonometrical identities.

• solve real life problems using trigonometircal ratios

V. STATISTICS AND

PROBABILITY

Statistical Data

• differentiate primary and secondary data

• collect data from their environment

• classify and tabulate primary data according to the required

criteria.

• construct a frequency distribution table for ungrouped data

• construct a histogram for a given data

• interprate a given histogram

• determine the Mean, Median and Mode of a given data

• describe the purposes and uses of Mean, Median and Mode

• identify the properties of the Mean of a given data (population

function)

• compute the measures of dispersion for ungrouped data

(manually and using scientific calculator)

• describe the purpose and use of measures of dispersion for


6


ungrouped data.

• determine the probability of an event from a repeated

experiment.

• determine the probability of an event.

VI. PLANE GEOMETRY

AND MEASUREMENT

• show that the sum of the measures of the interior angles of a

triangle is 1800

• find the measure of each interior angle of a regular polygon

• state properties of regular polygons.

• determine the lines of symmetry of regular polygons

• use the postulates and theorem on congruent triangle in

solving related problems.

• define similar plane figures and similar solid figures.

• apply the SSS, SAS and AA similarity theorems to prove

similarity of triangles

• discover the relationship between the perimeters of similar

plane figures and use this relationship to solve related

problems.

• discover the relationship between the areas of similar plane

figures and use this' relationship to solve related problems.

• discover the relationship between the volumes of similar

solid's and use this relationship to solve related problems.

• enlarge and reduce plane figures by a given scale factor.

• solve real life problems using the concepts of similarity and

congruency.

• describe radian measure of an angle.

• convert radian measure to degree measure and vice versa.

• use the trigonometrical ratios to solve right angled triangles.

• derive the distance formula (to find distance

between two points in the coordinate plane)

• apply the distance formula to solve related

problems in the coordinates plane

• determine the coordinates of points that divide a

given line segment in a given ratio

• define the gradient of a given line

• determine the gradient of a given line (given two

points on the line)

• determine the equation of a given line

• identify whether to lines are parallel or not.

• identify whether two lines are perpendicular or not.

• apply the properties of the slopes of parallel and

perpendicular lines to solve related problems

• apply the incidence theorems to solve related

problems.

• apply theorems on special quadrilateral in solving

related problems

• Apply the theorems on angles and arcs determined

by lines intersecting inside, on and outside a circle

to solve related problems

VI. PLANE GEOMETRY

AND MEASUREMENT

• find the angle whose trigonometrical value is given (using

trigonometrical table. )

• find the trigonemetrical values of angles from

trigonometrical table.

• determine the trigonometrical values for obtuse angles using

trigonometrical table.

• discover the symmetrical properties of circles

• use the symmetrical properties of circles to solve related

problems

• calculate the perimeters of regular polygons

• calculate the areas of regular polygons

• apply the formulae for calculating surface area and

volume of prism and cylinder

• calculate surface areas of a given pyramid or a

cone

• calculate the volumes of a given pyramid or a

cone.

• calculate the surface area of a given sphere


7


(cont.)

Vectors in Two Dimensions

• state angle properties of circles in their own words.

• apply angle properties of circles to solve related problems

• Find arc length, perimeters and areas of segments and sectors

• calculate areas of triangles using Heron's formula, whenever

the lengths of the three sides only are given.

• calculate areas of parallelograms.

• Calculate the surface areas of cylinders and prisms.

• Calculate volumes of cylinders and prisms

• differentiate Vectors from scalars quantities.

• represent vectors pictorially

• explain what is meant by magnitude and direction of a

vector.

• determine the sum of given vectors

• multiply a given vector by a given scales.

• express any given vector as position vector.

• calculate the volume of a given sphere

• define frustums of a pyramid and of a cone.

• calculate the surface areas of frustums of pyramids

of cones.

• calculate the volumes of pyramids or of cones.

• determine the surface area of simple composed

solids.

• calculate volumes of simple composed solids


8

Statement of Minimum Learning Competencies (MLCs) in Mathematics for Grade 11 & 12


I. NUMBER SYSTEM

The set of Complex Number

• add complex numbers correctly

• subtract complex numbers correctly.

• describe the closure property of both addition and

subtraction.

• describe the commutative and associative properties of

complex numbers.

• identify the additive identity element in �� .

• determine the additive inverse of a given complex number.

• determine the product of two complex numbers.

• describe five basic properties of multiplication of complex

numbers.

• divide two complex numbers

• give reason for each step in the process of division of

complex numbers

• determine the conjugate of a given complex number.

• find the Modulus of any given complex number

• Write the simplified form of expressions involving complex

numbers

• describe how to set up the Argand Plane.

• Plot the point corresponding to a given complex numbers.

• identify the complex number that corresponds to a given

point in the Argand Plane.

• represent any complex number in the polar form

• determine the modulus and argument of a given complex

number.

II. ALGEBRA

Rational Expression

• define rational expression

• identify the universal set of a given rational expression

• show the simplified form and the necessary steps in

simplify a given rational expression

• Perform the four fundamental operations on rational

expression


9


Matrices and Determinants

• decompose rational expressions into sums of partial

fractions.

• solve rational equations

• solve rational inequalities by using algebraic method (by

considering all possible cases)

• solve rational inequality by using the sign chart method

• define matrix

• determine the sum and difference of two given matrices of

the same order.

• multiply a matrix by a scalar

• describe the properties of multiplication of matrices by

scalars.

• determine the product of two matrices.

• describe the properties of the product of two matrices.

• determine the transpose of a matrix

• determine the determinant of a square matrix of order 2.

• determine the minor and cofactor of a given element of a

matrix

• calculate the determinate of a square matrix of order 3.

• describe the properties of determinants.

II. ALGEBRA (cont.)

Matrices and Determinants (cont.)

Introduction to Linear

Programming

• determine inverse of a square matrix

• find associated augmented matrix of equations

• describe elementary operations on matrices

• solve systems of equations in two or three variables using

the elementary operations

• apply Cramer's rule to solve systems of linear equations

��For social science stream only

• draw graphs of linear inequalities

y ≤ mx + c and

y ≥ mx + c and

ax + by ≤ c

• find maximum and minimum values of a given objective


10


function under given constraints.

• create inequalities from real life examples for linear

programming and solve the problem

II. ALGEBRA (cont.)

Mathematical Applications in

Business

�� For social science stream only ( cont.)

• compare quantities in terms of ratio.

• calculate the rate of increase and the rate of decrease in

price of commodities.

• solve problems on proportional variation in business

• solve problems on compound proportion

• find a required percentage of certain given amount

• compute problems on percentage increase or percentage

decrease

• calculate payment by installment for a given simple interest

arrangement.

• calculate the compound interest of a certain amount

invested for a given period of time.

• apply the formula for compound interest to solve practical

problems

• compute annuity for a give arrangement in compound

interest.

• describe what is depreciation mean and some its causes

• compute depreciation by using either of the two methods

appropriately.

• list five valid reasons for savings.

• explain how savings become investment.

• list three saving plans.

• identify four kinds of financial institutions.

• describe three main factors in choosing a particular

institution for saving.

• identify the four factors that should guide consumers in

planning an investment strategy.

• explain the differences between stocks and bond.

• describe ways to invest in stock and bond

�� For social science stream only

• find unit cost

• find the most economical purchase

• find total cost

• apply percent increase and percent decrease to

business

• apply percent increase and percent decrease to

business

• calculate initial expenses of buying a home

• calculate ongoing expenses of owing a home

• calculate commissions, total hourly wages, and

salaries

��For social science stream only ( cont.)


11


II. ALGEBRA(cont.)

Mathematical Applications in

Business

• describe the advantages and disadvantages of borrowing

money

• identify the usual sources of cash loan..

• compute the amount and time on the return of loan based on

the or given agreement.

• name three types of activities that government performs and

give examples of each

• explain why government collect taxes.

• describe the basic principles of taxation

• describe the various kinds of taxes.

• give their opinion about "income taxes" mean for them in

relation to their future first job.

• calculate different types of taxes based on the "rate of tax"

in Ethiopia

III. RELATION AND

FUNCTION

Further on Relation and

Function

• find out the inverse of a given relation

• Sketch the graph of a relation and its inverse.

• define power functions

• describe the properties of powers functions in relation to

their exponents

• determine the domains and ranges of power functions

• sketch the graphs of power functions

• define Modulus Function (Absolute value Function,

• determine the domain and the range of modulus function

• sketch the graph of a Modulus function

• define the Signum function

• determine the domain and range of Signum function

• sketch the graph of the Signum function

• define the "Greatest Integer Function"

• determine the domain and range of the Greatest Integer

function

• Sketch the graph of the Greatest Integer function

• define "one-to-one" function

• identify functions as one-to-one


12


• define "on to' function

• identify functions as on to

• identify one-to-one correspondence

• define the composition of function.

• determine the composite function given the component

functions

• determine the domain and the range of a composite function

of two given functions.

III. RELATION AND FUNCTION (cont.)

Further on Relation and

Function

• define inverse function

• describe the condition for the existence of inverse function

• determine inverse function for an invertible function.

• determine whether two given functions are inverses of each

other or not.

• Sketch the graph of the inverse of a function

• determine the domain and range of the inverse of a given

function

• define rational function.

• determine the domain of a given rational function.

• determine the range of a given rational function.

• sketch the graph of a given rational function

• determine the intercepts and symmetry of the graph of a

given rational function

• identify the type asymptote that the graph of a given

function may have.

• tell the properties of a given rational function from its

graph.

• use graphs of rational functions to solve rational inequalities

III. RELATION AND

FUNCTION (cont.)

Further on trigonometric

��For Natural Science stream only

• define and describe the functions sec x, cosec x and cot x.

• Sketch graphs of sec x, cosec x and cot x


13


functions • define and describe the functions sec x, cosec x and cot x.

• Sketch graphs of sec x, cosec x and cot x

• Sketch the graphs of

y = a sin x,

y = a sin kx;

y = a sin (kx +b),

y = a sin (kx +b) + c

• List the properties of these graphs.

• Sketch the graphs of

y = a cos x,

y = a cos kx

y = a cos (kx + b)

y = a cos (kx + b) + c

• List the properties of these graphs.

• Apply trigonometric functions to solve problems from fields

of science, navigation, engineering etc

III. RELATION AND

FUNCTION (cont.)

Sequences and Series

• revise the notion of sets and functions.

• explain the concepts sequence, term of a sequence,

rule (formula of a sequence)

• compute any term of a sequence using rule(formula).

• draw graphs of finite sequences.

• determine the sequence, use recurrence relations such

as, un+1= 2 un + 1,given u1

• generate the Fibonacci sequence and investigate its

uses, appearance in real life

• define arithmetic progressions and geometric

progressions.

• Determine the terms of arithmetic and geometric

sequences

• use the sigma notation for sums.

• compute partial sums of arithmetic and geometric

progressions

• apply partial sum formula to solve problems of

science and technology

• define a series


14


• decide whether a given geometric series is divergent

or convergent.

• show how infinite series can be divergent or

convergent

• show how recurring decimals converge

• discuss the applications of arithmetic and geometric

progressions (sequences) and series in science and

technology and daily life.

IV. LOGIC

Mathematical Reasoning

• explain the difference between "statement" and "open

statement"

• determine the truth value of a statement

• describe the rules for each of the five logical connectives.

• use the symbols ¬ , ∧ , ∨,⇒ and � to make compound

statements

• determine truth values of compound statements connected

by each of the logical connectives.

• determine truth values of two or three statements connected

by two or three connectives

• describe the properties and laws of logical connectives

• determine the equivalence of two statements

• define "Contradiction and "Tautology"

• determine that a given compound statement is either a

contradiction or tautology or neither of them

• find the "converse" of a given compound statement

• determine the truth value of the converse of a given

compound statement

• find the "contra -positive" of a given statement

• determine the truth value of the contra- positive of a given

statement

• recall what they have studied about statements and

logical connectives in the previous grade

• revise open statement

• understand the concept of quantifiers

• determine truth values of statements with quantifiers.

• define argument and validity

• check the validity of a given argument

• use rules of inference to demonstrate the validity of a

given argument

• distinguish between the nature of different types of

mathematical proofs.

• apply the right type of proof to solve the required

problem

• apply the principle of mathematical induction for

proving

• identify a problem and determine whether it could be

proved using principle of mathematical induction or

not.


15


• describe the two types of quantifiers

• determine the truth value of statements involving quantifiers

• describe what is meant by "argument"

• check the validity of a given argument

• use rules of inference to demonstrate the validity of a given

argument.

V. STATISTICS AND

PROBABILITY

Statistics and Probability

• identify qualitative and quantitative data

• describe the difference between discrete and continuous

variables (data)

• identify ungrouped and grouped data

• determine class interval (class size) as required to form

grouped data from a given ungrouped data

• make cumulative frequency table for grouped data (for both

discrete and continuous)

• described terms related to grouped continuous data, i.e.,

class limit, class boundary, class interval and class

midpoint.

• determine class limit, class boundary, class interval and

class midpoint for grouped continuous data.

• find the mean of a given grouped data.

• find median grouped discrete data

• find median for grouped data (continuous variable)

• determine the mode of a given grouped data.

• identify data that are unimodal, bimodal and multimodal.

• determine the quartiles for a given grouped data

• determine the required deciles of a given frequency

distribution

• determine the required percentile of a given frequency

distribution.

• describe the dispersion of data values

• find the range of a given data.

• Compute variance for ungrouped data

• ��For social science stream only

• describe the three methods/techniques of sampling.

• explain the advantages and limitation of each

techniques of sampling.

• describe the different ways of representations of data.

• explain the purpose of each representation of data.

• Construct graphs of statistical data

• identify statistical graph.

• explain the significance of representing a given data

in different types of graphs.

• draw histogram for a given frequency distribution

• Sketch frequency polygon for a given frequency

distribution

• sketch frequency curve for a given frequency

distribution

• draw bar chart

• construct line graph for data related to time.

• construct pie chart for a given data.

• compute the three mean divations of a given data.

• describe the relative significance of Mean divation as

a measure of dispersion.

• calculate the inter-quartile range for a given data.

• describe inter-quartile range as a measure of

variability in values of a given set of data.

• describe the usefulness of standard deviation in

interpreting the variability of a given data.


16


V. STATISTICS AND

PROBABILITY (cont.)

Statistics and Probability (cont.)

• calculate variance for grouped data.

• solve problems on variance

• Calculate standard deviation for grouped data.

• determine the number of different ways of possible

selections from a given sets of objects (by using the

multiplication principle)

• find the number of ways of selections of mutually exclusive

operations (by using the addition principle)

• determine the factorial of a given non-negative integer

• find the possible ways of arranging objects by using the

principle of permutation

• compute the possible arrangement of objects around the

circle (using the principle of circular permutation)

• describe the difference between arrangement of objects and

selection of objects.

• describe what is meant by "combination of objects"

• determine the number of different combinations of n

objects, taken r at a time.

• explain the computational relationship between

permutation and combination of objects.

• prove simple facts about combination.

• solve practical problems on combination of objects

• write up to the 6th power of a binomial expression (x + y)

n

(i.e. when n = 0, 1, 2, 3, 4, 5 ) in its expanded form by using

direct multiplication

• describe what they observe in the expansion of (x + y)n

where n = 0, 1,2,3,4,5

• describe "Pascal’s Triangle" and its use

• apply the "Binomial Theorem" in expanding the nth power

of binomial terms i.e. (x + y)n, where n∈Z

+

• determine any term in the expanded form of (x + y)n where

n∈ Z+

solve problems on binomial expansion

��For social science stream only ( cont. )

• compare two groups of similar data..

• determine the consistency of two similar group of

data with equal mean but different standard

deviations

• describe the application of coefficient of variation inn

comparing two groups of similar data.

• describe the relationship among mean, median and

mode for grouped data by using its frequency curve.

• use cumulative frequency graphs to determine the

dispersion of values of data (interms of its Mean,

Median and Standard deviation)

• determine the variability of value of data interms of

quartiles by using cumulative frequency graph.

• describe the relationship among mean, median and

mode for grouped data by using its frequency curve.

V. STATISTICS AND PROBABILITY (cont.)

• determine any term in the expanded form of (x + y)n ,where

n∈Z+

• solve problems on binomial expansion

• describe what is meant by "Random Experiment"

• explain what is meant by an outcome of a random


17



experienced

• describe what is meant by sample space of a given random

experiment.

• list some of the sample points of a sample space for a given

experiment.

• define "equally likely outcomes" of a given trial in his own

words.

• define "favorable outcomes/ cases"

• determine events of a given random experiment

• identify sample (elementary) events and compound events

• determine the number of events of a given sample space

• describe the occurrence or non occurrence of an event.

• explain an event denoted by "not E" where "E" is a given

event

• explain events connected by "or" and "and"

• describe the simplified forms of events by using the

properties of operations on sets

• identify exhaustive events

• identify mutually exclusive events

• describe events that are both exhaustive and mutually

exclusive

• identify independent events.

• identify dependent events

• describe the axiomatic approach of probability

• interpret basic facts in the theory of probability.

V. STATISTICS AND PROBABILITY (cont.)


• find probabilities of events based on

• find probabilities of events based on “Axiomatic” approach.

• describe the odds infamous of an event or the odds against

an event

• Find the probability of E1 ∪ E2 where E1 and E2 are

events in a random experiment

• determine the probability of mutually exclusive events.

• find probability of the joint occurrence independent event

(by using rule of multiplication)

• describe the out comes of events using tree diagram

• determine the probability of the joint occurrence of


18


dependent events (using multiplication rule)

• describe the outcomes of events using tree diagram to

determine their probability

• identify whether a given events are independent or

dependent (by comparing the equation for probability of

joint occurrence of independent events).

VI. CALCULUS

Limits of sequence of numbers

• define upper and lower bound of number sequences.

• find out the least upper (greatest lower) bound of

sequences.

• define limit of a number sequence

• consolidate their knowledge on the concept of

sequences stressing on the concept of null sequence.

• apply theorems on the convergence of bounded

sequences

• prove theorem about the limit of the sum of two

convergent sequences.

• apply theorems on the limit of the difference,

product, quotient of two convergent sequences

• define limit of a function.

• determine the limit of a given function at a point.

• find out the limit of the sum, difference, product and

quotient of two functions.

• define continuity of a function in interval.

• describe the properties of continuous functions.

• use properties of continuous functions to determine

the continuity of various functions.

• consolidate what they have studied on limits.

• solve problems on limit and continuity to stabilize

what have learnt in the unit.

VI. CALCULUS

• find the rate of change of one quantity with respect to

another.

• sketch different straight line and curved graphs and

find out slopes at different points of each graph.

• define differentiability of a function at a point x 0 .

n→ ∞


19


Introduction to Differential

Calculus

Application of Differential

Calculus

• explain the geometrical and mechanical meaning of

derivative.

• set up the equation of tangent line at the point of

tangency, using the concept of derivative.

• find the derivative of elementary functions over an

interval.

• find the derivatives of power, simple trigonometric,

exponential and logarithmic functions

• apply the sum and difference formulae of

differentiation of functions.

• apply the product and quotient formulae of

differentiation of functions.

• apply the chain rule and differentiate composition of

functions

• find the 2nd

and the nth derivative of a function.

• consolidate and stabilize what has been studied in the

unit.

• consolidate the concept zero(s) of a function.

• find critical numbers and maximum and minimum

values of a function on a closed interval.

• explain the geometric interpretations of Rolle's

theorem and mean value theorem

• find numbers that satisfy the conclusions of mean

value theorem and Rolle's theorem.

• Solve problems on application of differential calculus

• Interpret and apply differential calculus on problems

involving the rate of change.

• consolidate what has been learnt in this unit


20


Introduction to Integral

Calculus

• differentiate between the concepts differentiation and

integration

• use the properties of indefinite integrates in solving

problems of integration

• integrate simple trigonometric functions

• use different techniques of integration for

computation of integrals

• Compute area under a curve.

• use the concept of definite integral to calculate the

area under a curve.

• state fundamental theorem of calculus

• apply fundamental theorem of calculus to solve

integration problems.

• state the properties of definite integrals.

• apply the properties of definite integrals for

computations of integration

• apply the knowledge on integral calculus to solve

problems.

VII. GEOMETRY

Coordinate Geometry

and Vectors

• write different forms of equation of a line.

• determine the slope, x-intercept and y-intercept of a line

from its equation

• determine the angle between two intersecting lines on the

coordinate plane whose equations are given.

• determine the distance between a point and a line given on

the coordinates plane.

• name the different types of conic sections

• explain how the conic sections are generated (formed)

• define circle as a locus and write equation of a circle

• find the radius and center of a circle from its equation.

• determine whether a given line and circle have a point of

intersection .

• determine the coordinates for the intersection point(s) (if the

given line and the given circle intersect)

• write equation of a tangent line to a given circle. (where the

point of tangency is given)

��For social science stream only

• construct the coordinate axes in space

• identify planes determined by the axes in space.

• identify the octants determined by the planes and

axes.

• read the coordinates of a point in space.

• describe how to locate a point in space.

• plot a point whose coordinates are given.

• give the equations for the planes determined by the

axes.

• show graphically how to find the distance between

two points in space.

• compute distance between two given points in space.

• determine coordinates of the mid-point of a segment

in space.

• describe the equation of a sphere

• derive equation of a sphere


21


• Write the standard form of equation of a parabola.

• draw different types of a parabolas

• describe some properties of a given parabola.

• define "ellipse" as a locus (set of points on the plane which

satisfy a certain given condition)

• write the standard form of equation of an ellipse and sketch

ellipse

• describe some terms related to ellipses ( such as latus

rectum, eccentricity, major and minor axes...)

• define hyperbola as a locus

• write the standard form of equation of an ellipse

• describe related terms to hyperbola (foci, centre, transverse

axis, asymptotes, conjugate axis...)

• sketch hyperbola based on its given equation

• give eccentricity of a given hyperbola solve problems on

hyperbola

• solve problems related with sphere

• add, subtract vectors and multiply by a scalar in space

• use the unit vectors i, j and k while representing a

vector.

• describe the properties of addition to solve exercise

problems..

• show the closure property on their own

• find the length of a vector in space

• find the scalar product of two vectors in space.

• evaluate and show the angle between two vectors in

space.

minimum learning competencies - info.moe.gov.etinfo.moe.gov.et/curdocs/mlcmath9-12.pdf · iv....

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